Physica D 152–153 (2001) 434–450

On the Badulin, Kharif and Shrira model of resonant water waves

Walter Craig1

Department of Mathematics, Brown University, Providence, RI 02912, USA

Dedicated to Vladimir Zakharov on his 60th birthday, with best wishes for many more to follow

Abstract

This paper is a reappraisal of the Hamiltonian model derived by Shrira, Badulin and Kharif (BKS) for three-dimensionalnonlinear water waves. The model was introduced in [J. Fluid Mech. 318 (1996) 375] in an effort to describe the formationof traveling waves with crescent-shaped features that arise from the instability of the Stokes wave train at moderately largesteepness. There have been observations of such traveling waves in wave tank experiments by Su et al. [J. Fluid Mech. 124(1982) 45–72] and Su [J. Fluid Mech. 124 (1982) 73–108]. Some of the regimes described in these papers are of lightlybreaking waves, which are asymmetric, with all crescents facing forward. Other regimes that they observe apparently give riseto traveling waves which have asymmetric crescent-shaped features facing both forwards and backwards. We show that the BKSmodel describes the Stokes wave train and its loss of stability at moderate amplitudes as a Hamiltonian saddle-node bifurcation,which corresponds to the formation of a stable three-dimensional wave pattern which exhibits asymmetric crescent-shapedelements. The model also produces a family of solutions homoclinic to the unstable Stokes wave train, which surrounds theorbit of crescent-shaped wave patterns and which provides a mechanism for transition. Other traveling wave solutions of theBKS model having nonzero transverse momentum are good candidates for the skew wave patterns possessing characteristichexagonal shaped structures separated by quiescent stripes which are produced to the sides of the experiments in wave tanks.The BKS model has solutions which satisfy two of the three characteristics specified in [J. Fluid Mech. 318 (1996) 375] fornonlinear crescent-shaped waves, avoiding the introduction of a dissipative mechanism to describe features of these familiarwave patterns. The one weakness of the BKS model is that the crescent-shaped wave patterns are transformed to themselvesunder time reversal composed with a phase shift. Therefore all of the wave patterns described by the BKS model possessforward and backward facing crescent-shaped elements simultaneously, associated with alternating crests. These solutionsreproduce the features of some but not all of the wave patterns in the observations of Su et al. [J. Fluid Mech. 124 (1982)45–72] and Su [J. Fluid Mech. 124 (1982) 73–108]. In the deep water case, we introduce and analyze a new and more realisticfour degrees of freedom Hamiltonian model of water waves which has two principal five wave interactions. While being morecomplicated and not completely integrable, nonetheless this model has traveling wave solutions with similar crescent-shapedelements, and others with the hexagonal features of the BKS model. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Resonant water waves; BKS model; Stokes waves

1. Introduction

Surface water waves are readily and informally observable in our common experience, and are a basic phenomenonwhich has been central to the study of nonlinear wave phenomenon for over a century. Two types of three-dimensional

E-mail address: craigw@math.brown.edu (W. Craig).1 Research partially supported by the National Science Foundation under grant #DMS 9706273.

0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0 1 6 7 -2 7 89 (01 )00184 -1

W. Craig / Physica D 152–153 (2001) 434–450 435

wave forms which appear among water waves are apparently ubiquitous; forward facing crescent-shaped steep waterwave patterns, and hexagonal, often large aspect ratio, traveling wave patterns. This paper reports on a model ofthree-dimensional water wave evolution that was introduced by Shrira et al. [21] in an effort to describe severalphenomena observed in wave tank experiments performed by Su et al. [23] and Su [24]. One of the observations thatwas reported in these papers was a class of regular and doubly periodic patterns of three-dimensional crescent-shapedwaves. The most striking of these is of lightly breaking waves, with all of the crescent-shaped features facingforwards. These solutions are apparently the result of a three-dimensional instability of the two-dimensional Stokeswave, which occurs at moderately large steepness. Indeed, essentially two-dimensional Stokes wave trains ofsufficiently large wave steepness which were initiated at the paddle of the experimental wave tank facility, developafter a number of basic periods into this three-dimensional pattern of crescent-shaped waves. The three-dimensionalpattern persists for a substantial number of basic periods before radiating a skew wave pattern to the lateral sidesof the wave tank, and returning to a lower amplitude (and relatively noisy) approximately two-dimensional finalstate. Much of the literature which discusses this experiment has concentrated on the crescent-shaped pattern, whilewe note that the second paper of Su [24] focuses on the skew wave patterns. In the present paper, we present andanalyze a model of nonlinear water waves which exhibits solutions with both classes of behavior.

Theoretical descriptions of the experiments of Su et al. start with the numerical solutions of Meiron et al. [16],who note that the experiments are close to a bifurcation phenomenon for three-dimensional traveling water wavepatterns. Their paper gives a number of numerical calculations of three-dimensional solutions of the water waveequations, both in Eulerian coordinates and in Zakharov’s Hamiltonian formulation of the problem [25]. Theirresults give consistent agreement between the two choices of coordinates, at approximately the observed wavesteepness. However, their solutions are in all cases symmetric under reflection through a plane at each wave crestorthogonal to the direction of propagation, and the crescent-shaped features that they exhibit appear in both forwardand backward orientation simultaneously. The subsequent paper of Saffman and Yuen [20] is cogniscent of the roleof a basic five wave interaction in the formation of these wave patterns, but the solutions that they describe remainsymmetric about each crest, and furthermore no stability argument is presented.

The present paper is based on a new idea that was introduced by Shrira et al. [21]. They develop the elegant pointof view of Zakharov that describes water wave evolution as a Hamiltonian system with infinitely many degreesof freedom. Starting from Zakharov’s Hamiltonian formulation for Euler’s equations [25], and posing the basicequations in complex symplectic coordinates, Badulin, Kharif and Shrira (BKS) first subject the system to severalcanonical transformations (renormalization transformations) to achieve a normal form. Their model describingthree-dimensional crescent wave formation is derived directly from the Hamiltonian in fifth-order Birkhoff normalform, which is then truncated to include only the evolution of the complex amplitudes for a principal Fourier wavenumber k(1), and two side modes k(2) and k(3) in symmetric position, which are in a fifth-order resonance with k(1).BKS assume that most of the basic phenomena of nonlinear traveling water waves are described by the interactionsbetween these three complex Fourier modes.

However, after a preliminary analysis of this model, BKS concluded that crescent patterns did not occur in theHamiltonian equations for water waves, and that dissipation at some scale was crucial for their formation. Thispaper [21] introduces two dissipative modifications of their original model for this purpose. It is not unreasonableto speculate that dissipation may play a role, as some of the observations of Su et al. [23] and Su [24] are ofcrescent-shaped patterns on waves which are lightly breaking. However, the thesis of the present paper is thatthree-dimensional crescent-shaped waves occur in smooth free surface flows described by the Hamiltonian modelof BKS itself. Indeed, a more in-depth look at this Hamiltonian model reveals that a number of classes of travelingwave solutions exist, some of them exhibiting relatively large steepness and strong periodic crescent-shaped featuresthat are some of the hallmarks of the observed wave patterns of Su et al. [23] and Su [24]. These solutions are ofpermanent form, and in particular they exist without the presence of breaking or other mechanisms of dissipation.

436 W. Craig / Physica D 152–153 (2001) 434–450

Additionally, there are time dependent solutions of the model which are homoclinic to a two-dimensional travelingwave pattern. These solutions start very near to the two-dimensional traveling wave, and then evolve to passnearby to a crescent-shaped wave of permanent form, remaining in a neighborhood over many basic periods beforereturning asymptotically to the two-dimensional traveling wave. Recent observations by Collard and Caulliez [2]of wave patterns in a wave tank at Luminy, which are formed of crescent-shaped waves which are not breaking,are consistent with solutions having these properties. Additionally, we present a class of solutions with nonzerotransverse momentum which are candidates for the skew wave patterns also described in the observations of Su et al.[23] and Su [24]. These have a strong tendency toward a (nonsymmetric) hexagonal form which is characteristic ofwave patterns which are principally composed of two Fourier components with wavenumbers at an acute angle to eachother, and they are related to the observations and KP modeling of Hammack et al. [10,11] in nonresonant settings.

Our principal findings for the Hamiltonian BKS model and the four mode model of resonant water waves are asfollows. (i) It is a completely integrable, three degrees of freedom Hamiltonian system. The integrals can be taken tobe the two components of the horizontal momentum I1 and I2, a well as the total energy HBKS itself. Action–anglevariables correspond to rotating coordinates, in which stationary points correspond to traveling wave solutions.(ii) For transverse momentum I2 = 0 the model possesses a solution ηS(x− tc) analogous to the Stokes wave train,a nonlinear two-dimensional traveling wave solution, and this solution is a stable elliptic orbit up to a specific valueof the momentum I1 = b1. (iii) When I1 exceeds the value b1, the two-dimensional solution ηS(x − tc) undergoesa Hamiltonian saddle-node bifurcation, loosing its stability and acquiring one pair of hyperbolic directions, andan orbit homoclinic to ηS(x − tc) is formed. A new stationary point ηC(x − tc) is created, which is elliptic andtherefore stable, corresponding to a traveling wave with three-dimensional character. The wave pattern describedby ηC(x − tc) occurs with a phase difference of π/3 between the central Fourier mode and the smaller amplitudeside bands, and the resulting traveling wave has the character of a crescent-shaped pattern. The homoclinic orbitsurrounds this stationary point ηC(x − tc), and for momentum I1 not too much larger than b1, this orbit spendsa time equivalent to numerous basic periods in a neighborhood of the elliptic stationary point. It thus exhibits arobust tendency to mimic the crescent-shaped patterns of the elliptic stationary point ηC(x − tc) for appreciablelengths of time, before returning asymptotically to the two-dimensional traveling wave. (iv) For the same values ofthe momentum, the model has traveling wave solutions ηH (x − tc) which are hexagonal in structure, having thecharacter of a nonlinear superposition of 2 two-dimensional wave trains of equal amplitudes intersecting at an obliqueangle. Similar solutions would also exist for systems out of resonance, and are related to the work of Hammacket al. [10,11] in the long wave regime as described by the KP equation. (v) For transverse momentum I2 �= 0,the model exhibits traveling wave solutions which give rise to (generally nonsymmetric) hexagonal wave patterns.We propose these latter solutions as candidates for the skew wave patterns observed by Su [24]. The characteristicstripes occurring in these patterns which run at an oblique angle to their phase velocity vector correspond to raysdrawn through the smaller amplitude side bars of the hexagonal pattern, and these can be clearly seen in numericalsolutions of the full Euler equations [18].

One question as to the accuracy of the BKS Hamiltonian model has to do with the process of approximation bymode truncation. In the case of infinitely deep water we provide a partial resolution of this by employing a newmodel of resonant water waves in which one additional mode is included, which is involved with the modes of theBKS model via a second fifth-order resonant interaction. Although more complicated, the principal features of thetraveling wave solutions of this new model are roughly the same as for the BKS model, and we obtain analogousexistence and stability results.

There is one principal weakness of the solutions produced by both the BKS model and the new model of resonantwater waves, which has to do with the crescent-shaped elements of solutions, and the fact that in actual water wavesthe most striking patterns have crescents which are all facing in the direction of propagation of the waves. The waterwave equations are reversible in time, so that every forward facing solution profile has to be matched by another

W. Craig / Physica D 152–153 (2001) 434–450 437

one that is facing backwards. The one that appears in physical experiments should be determined by a criterion ofstability. In the BKS model, the class of solutions which are good candidates for crescent-shaped waves are stable,and do not have crests which are symmetric under reflection in a plane perpendicular to the direction of propagation.However, they do exhibit a symmetry in that time reversal, composed with a spatial translation, transforms one ofthese solutions to itself. In particular, a solution of the BKS model which possesses forward facing crescent-shapedfeatures also simultaneously possesses identical but backwards facing crescent-shaped features from other crests.These patterns are similar to those seen in some of the observations reported in [23,24], but they do not reproducethe observed wave patterns in which the crescents are principally facing forwards. Nevertheless, these solutions ofthe two resonant five wave models of the water wave problem exhibit two of the three principal features that arespecified in [21] as hallmarks of nonlinear crescent-shaped waves, without resorting to a mechanism of dissipation.It remains an open question whether a Hamiltonian model can also satisfy the third condition of Shrira et al. [21],for all crescent-shaped elements to be facing the same direction.

Using the predictions of these resonant water wave models, accurate three-dimensional traveling water wavecalculations by Craig and Nicholls are presently under way, using spectral codes which provide convergence criteriawith spectral accuracy to traveling wave solutions of the full Euler equations. These will be described in detail inthe separate publication [18].

The analysis by means of a Birkhoff normal form for a Hamiltonian system is consistent with the spirit ofZakharov’s elegant program, where one understands nonlinear phenomena in partial differential equations as be-ing manifestations of analogues of Hamiltonian mechanics in the presence of infinitely many degrees of freedom.Zakharov’s work has been enormously influential over the past number of decades, as is testified by the breadth andthe variety of articles in this volume dedicated to him on his 60th birthday. I am personally grateful for what he hastaught me.

2. Hamiltonian description of the water wave problem

The Euler equations for an ideal fluid in three dimensions with a free surface are the starting point in this problem,and the BKS model equations are derived from them, mostly along principles of classical Hamiltonian mechanics.In a fluid domain S(η) = {(x1, x2, y) ∈ R3, t ∈ R : −h < y < η(x, t)}, the velocity field is given in Euleriancoordinates by the gradient of a potential function ϕ which satisfies

�ϕ = 0 for − h < y < η(x, t), ∂yϕ = 0 at y = −h,∂tη = ∂yϕ − ∇xϕ · ∇xη, ∂tϕ = −gη − 1

2 |∇ϕ|2 at y = η(x, t), (1)

where g is the acceleration of gravity and h the average depth of the fluid. It is possible to have h = +∞. We willimpose periodic boundary over a fundamental domain T (Γ ) = R2/Γ , where Γ ⊆ R2 is a lattice.

This problem can be written in Hamiltonian form, using the Hamiltonian functional and choice of canonicalvariables that was originally proposed by Zakharov [25]

HZ = 1

2

∫T (Γ )

∫ η(x)

−h(∇ϕ(x, y))2 dy dx + g

2

∫T (Γ )

η2(x) dx. (2)

Introducing the canonically conjugate variables (η(x), ξ(x) = ϕ(x, η(x))) as in [25], the Hamiltonian (2) can berewritten in terms of (η, ξ) in explicit form using the Dirichlet–Neumann operatorG(η), as described by Craig andSulem [5]

H(η, ξ) =∫T (Γ )

1

2ξG(η)ξ + g

2η2 dx, (3)

438 W. Craig / Physica D 152–153 (2001) 434–450

and the system (1) of partial differential equations is equivalent to the Hamiltonian system

∂t

(η

ξ

)=(

0 I

−I 0

)(∂ηH

∂ξH

). (4)

We will denote the flow of (4) by Φt(η, ξ). Since the Hamiltonian (3) is a quadratic form in ξ , one vector fieldcomponent is ∂ξH = G(η)ξ . The other component ∂ηH is more interesting, being the variation of the Dirichletintegral with respect to the domain defined by the graph of η, and the computation is related to the Hadamardvariational formula [9] for the Green’s function of the fluid domain S(η). After a calculation which can be found in[4], system (4) has the form

∂tη = G(η)ξ,

∂t ξ = −gη − 1

2(1 + |∇xη|2) (|∇xξ |2 − (G(η)ξ)2 − 2(G(η)ξ)∇xξ · ∇xη + |∇xξ |2|∇xη|2 − (∇xξ · ∇xη)2).

(5)

The Dirichlet–Neumann operator G(η) maps the Hilbert spaces H 1(T (Γ )) to L2(T (Γ )), and is analytic in itsdependence upon η ∈ C1(T (Γ )), therefore we may write it as a convergent Taylor series expansion G(η) =G0+

∑∞m=1Gm(η), whereGm(η) is homogeneous of degreem inη. In parallel, the Hamiltonian (3) has a well-defined

Taylor expansion

H(η, ξ) = H2(η, ξ)+∑m≥3

Hm(η, ξ) =∫T (Γ )

1

2ξG0ξ + g

2η2 dx +

∑m≥3

∫T (Γ )

1

2ξGm−2(η)ξ dx. (6)

The linearized equations come from the quadratic term H2 of the Hamiltonian, in which G0 = |D| tanh(h|D|),using the definition D = −i∂x . From this expression, we obtain the well-known linear dispersion relation

ω2(k) = g|k| tanh(h|k|). (7)

The linear Fourier modes which satisfy the conditions of periodicity over the fundamental domain T (Γ ) = R2/Γ

have wavenumbers k in the dual lattice Γ ′, and these have the temporal frequencies ω(k), k ∈ Γ ′.Express the canonical variables (η, ξ) in their Fourier series expansion

η(x) =∑k∈Γ ′

η(k) eik·x, ξ(x) =∑k∈Γ ′

ξ (k) eik·x (8)

with the reality condition that η(k) = η(−k), ξ (k) = ξ (−k). We introduce the complex symplectic coordinates

z(k) =√

g

2ω(k)η(k)+ i

√ω(k)

2gξ(k), k ∈ Γ ′ (9)

in which the Hamiltonian has the form

H(z, z) =∑k∈Γ ′

ω(k)|z(k)|2 +∑m≥3

∑

|P |+|Q|=mc(P,Q)zP zQ

, (10)

where P = (p(k))k∈Γ ′ , Q = (q(k))k∈Γ ′ : Γ ′ → N are multi-indices with p(k), q(k) ∈ N as the individualexponents of the Taylor monomials, with zP = ∏

k∈Γ ′zp(k)(k) and zQ = ∏k∈Γ ′ zq(k)(k) forming the Taylor

W. Craig / Physica D 152–153 (2001) 434–450 439

monomials themselves, of degree |P | = ∑k∈Γ ′p(k) times |Q| = ∑

k∈Γ ′q(k), and the coefficients c(P,Q) arederived from (6), (8) and (9) in the usual way. In complex symplectic coordinates, system (4) is expressed as

∂t z(k) = i∂z(k)H(z, z). (11)

The flow Φt of Eq. (10) conserves the two components of the horizontal momentum vector (I1, I2), which is to saythat {H, Ij } = 0 for j = 1, 2. As a consequence, we have the following easily verified result.

Proposition 1. The only coefficients c(P,Q) which are nonzero satisfy the two conditions

〈P −Q, k〉 =∑k∈Γ ′

(p(k)− q(k))k = 0 (12)

for k = (k1, k2) ∈ Γ ′.

Another basic conserved quantity is M = ∫T (Γ )

η dx = |T (Γ )|1/2η(0), which is the added mass. Indeed onecan check that {M,H } = 0, and therefore we may and will restrict our considerations of the evolution of (4) tothe subspace of Fourier series with η(0) = 0. In a real situation this simply corresponds to a proper choice of theparameter h.

A monomial term c(P,Q)zP zQ in the HamiltonianH in (10) is said to be resonant of orderm if its multi-indices(P,Q) satisfy

〈P −Q,ω〉 =∑k∈Γ ′

(p(k)− q(k))ω(k) = 0, |P | + |Q| = m (13)

with ω = (ω(k))k∈Γ ′ . This is of course to be taken in conjunction with the conditions (12) in order to be relevant.Resonant triads. When |P | + |Q| = 3, there are no solutions of the three equations (12) and (13), as a simple

geometrical argument with the dispersion relations will show [6]. If the effects of surface tension are included, thenthis conclusion is no longer true, and resonances are possible.

Resonant quartets. When |P | + |Q| = 4, the three conditions (12) and (13) for resonance do have solutions. Thegeneric resonances P = Q are of course present, although the associated resonant monomials are quadratic in theaction variables I (k) = |z(k)|2, and are therefore benign. A number of other resonant terms are possible, dependingin a sensitive way on the depth h. In case h = +∞, and considering only two-dimensional fluid motions, there areBenjamin–Feir resonances, corresponding to the resonant monomials of the Hamiltonian H of degree 4 which aredescribed by the multi-indices

P −Q = δqm2 − δ−q(m+1)2 − δ−qm2(m+1)2 + δ−q(m2+m+1)2 ,

where δ* is the multi-index with all zero entries except for that of index *, where it is 1. For each q ∈ N, m ∈ Nthis reflects a four wave resonant interaction between the wave numbers k(1) = qm2, k(2) = q(m + 1)2, k(3) =qm2(m+1)2, and k(4) = q(m2+m+1)2, that is, anm : m+1 : m(m+1) : m2+m+1 relationship of proportionalitybetween the temporal frequencies. It has turned out, however, that these are irrelevant to any dynamical questionsof deep water waves, as the associated coefficients c(P,Q) vanish in a Birkhoff normal forms calculation [6,7].In the two-dimensional problem with h = +∞, the two above-mentioned possibilities are the only two classes ofresonant quartets. In cases where 0 < h < +∞, or in three-dimensional cases, there are other possible resonantquartets which can occur, again depending upon h and the fundamental domain T (Γ ) in a sensitive manner.

Resonant quintets. In the case of two-dimensional fluid motion and for h = +∞, the class of five wave resonantinteractions has been characterized in [6,8,15], and in the latter reference the Birkhoff normal for has been expressedin full. The simplest resonant term corresponds to a fifth-order 1:2 resonance, with P −Q = 3δm − δ−m − δ4m for

440 W. Craig / Physica D 152–153 (2001) 434–450

any m ∈ N. In [3], it is shown that this term generally leads to an instability of any sufficiently accurate finite modeapproximation to the water wave problem. When 0 < h < +∞ there can be many possible resonant quintets.

In the case of fully three-dimensional fluid motions, the most important resonant quintet is directly related tothe problem of persistent crescent-shaped wave patterns. The proposal for this originates in the papers of Su et al.[23] and Su [24], see also [17,19]. The resonance stems from one principal wave number k(1) = (k

(1)1 , 0) and its

interactions with two satellites k(2) = (k(2)1 , k

(2)2 ) = ( 3

2k(1)1 , k

(2)2 ) and k(3) = (k

(3)1 , k

(3)2 ) = ( 3

2k(1)1 ,−k(2)2 ). By

construction 3k(1) − k(2) − k(3) = 0, and the monomial z3(k(1))z(k(2))z(k(3)) will be resonant when

3ω(k(1))− ω(k(2))− ω(k(3)) = 0. (14)

Given a triple (k(1), k(2), k(3)) of wavenumbers satisfying (12) and (14), their integer linear combinations span aresonant lattice Γ ′

r .

Proposition 2. For each 0 < h ≤ +∞, there is a unique choice of the wavenumber component k(2)2 for which (14)

is satisfied. When h = +∞ then k(2)2 = ( 34

√5)k(1)1 , corresponding to an opening angle of Θ∞ = arctan( 1

2

√5)

between k(1) and k(2) or k(3). This resonant opening angle decreases monotonically to zero for decreasing depth h.

Proof. Since |k(2)| = |k(3)|, then ω(k(2)) = ω(k(3)). The RHS of (14) is continuous and monotone decreasing ink(2)2 , it is positive for k(2)2 = 0 by the concavity of the dispersion relation (7) in |k|, while it is negative for large k(2)2 ,

hence there is a unique root. When h = +∞ then ω(k) = √g|k|, therefore (14) reads

3√k(1)1 − 2

4√(k(2)1 )2 + (k

(2)2 )2 = 0.

Substituting k(2)1 = 32k

(1)1 and solving the resulting quartic for k(2)2 in terms of k(1)1 , one obtains the value for Θ∞ in

the proposition. In the limit of small h, the expression limh→0 ω(k)/h is linear in |k| and (14) is satisfied only fork(2) parallel to k(1). Furthermore, using (14), we have

3∂hω(k(1))− 2∂hω(k

(2))d

dhk(2)2 = 0 (15)

from which we deduce that (d/dh)k(2)2 = 32∂hω(k

(1))/∂hω(k(2)). Since ∂hω > 0 for nonzero |k|, the monotonicity

of Θ in h follows. �

The monotonicity in h of the opening angle of this resonant fundamental domain gives an amusing suggestionfor a nonlinear inverse problem to sample the depth under certain conditions, without knowing the wave height,steepness or phase velocity (but using information that is available from, for example, an aerial photograph). Givena wavefield which is essentially two-dimensional but undergoing a transition to a three-dimensional configuration,the two-dimensional period and the principal period exhibited in the third transverse direction define an openingangle for a fundamental domain T (Γ ). With the assumption that the most obvious instability arises from the aboveresonance, one then determines the (unique) average depth h such that (14) holds. This is a quantitative versionof the observation that three-dimensional structure of crests in open water occurs with relatively short transversewavelength, while near the shore the three-dimensional structure of crests tends to be widely spaced.

3. Normal forms transformations

An idea that has been often used in the subject of Hamiltonian evolution equations is that of normal forms;one performs a (finite) sequence of canonical transformations of the given system in order to reduce it to a linear

W. Craig / Physica D 152–153 (2001) 434–450 441

problem, or at least to a problem in the simplest nonlinear form. In the water waves problem, the quiescent statez = 0 is an elliptic equilibrium of the system, and the relevant goal is a Birkhoff normal form. At the mth stepof this procedure, nonresonant terms of the mth term of the Taylor expansion about z = 0 of the Hamiltonian Hare transformed to zero, with a canonical transformation which leaves the terms of order less than m invariant, andwhich possibly modifies terms of order m + 1 and higher. The formal aspects of this process lead us to expect atthe mth step, m = 3, 4, . . . , a near-identity canonical transformation f (m) which removes the nonresonant termsof the mth order Taylor polynomial of H(m−1) = H ◦ f (m−1) ◦ f (m−2) ◦ · · · ◦ f (3), leaving invariant the terms ofH(m−1) of its Taylor polynomials of orders j = 3, . . . , m − 1, and possibly modifying terms of order m + 1 orhigher. After the mth transformation the resulting Hamiltonian H(m) has the form

H(m)(z, z) = H2(z, z)+m∑j=3

H resonantj (z, z)+ R(m+1)(z, z) (16)

with

H resonantj =

∑P,Q:|P |+|Q|=j,

〈P−Q,ω〉=0

〈P−Q,k〉=0

c(j)(P ,Q)zP zQ. (17)

Unless the system is completely integrable in the action–angle variables (I (k), θ(k) = arg(z(k))) it is unlikely thatthe sequence of transformations f (m) ◦ f (m−1) ◦ · · · ◦ f (3) converges as m goes to infinity on any neighborhoodof the origin z = 0. Furthermore, Hamiltonian systems with infinitely many degrees of freedom, such as PDE andmore pertinently the water wave problem, have analytic issues associated with the individual transformations f (m),similar to the small divisor problem, which arise in their construction at each step of the process. The analysis ofthe Birkhoff normal form for the two-dimensional case of the transformation f (3) is described in [3], but not muchmore is known in general. We will only consider formal aspects of the Birkhoff normal forms transformations inthis paper. Discussions of normal forms transformations on a formal level for the water waves problem appear in[1,12]. Related considerations (in the continuum rather than in the periodic setting) appear in [26].

4. The model of BKS

Natural model problems for Hamiltonian systems which can be written in the form of (4), with Hamiltonian inBirkoff normal form to some order, are given by simply truncating the normal forms Hamiltonian

H(z, z) = H2(z, z)+m∑j=3

H resonantj (z, z).

In many cases of interest this, however, remains an infinite dimensional system in its own right, and to effectivelyobtain qualitative information it may be expedient to restrict the infinite-dimensional problem to a finite number ofvariables. That is, one fixes all but finitely many Fourier modes to be z(k) = 0, projecting the problem onto a finitedimensional subspace of the full phase space EM = {z(k) = 0, k �= k* : * = 1, . . . ,M}. This method of derivingmodel problems for the water waves problem by restriction to finite subspaces of modes appears in the work ofStiassnie and Shemer [22], however, without the Birkhoff normal forms transformations. We emphasize the factthat the subspace EM is rarely left invariant by the flow Φt(z, z) of the full problem, nor does the projection ontoEM usually commute with the flow Φt(z, z), so that this truncation is not necessarily a mathematically justifiableapproximation.

442 W. Craig / Physica D 152–153 (2001) 434–450

The derivation of the BKS model of Shrira et al. [21] follows the recipe above, performing a formalm = 5, Birkhoffnormal forms transformation on the three-dimensional water waves Hamiltonian (see [13,14]) and then restrictingthe resulting Hamiltonian to the subspace spanned by the Fourier modes with wave numbers k(1), k(2) and k(3) ofthe resonant lattice Γ ′

r . In [21], the authors set h = +∞ although this is not necessary as long as one can performthe sometimes tedious calculations for the normal form Hamiltonians H(m). The result from Shrira et al. [21] is

HBKS =3∑*=1

ω(k*)|z(k(*))|2 + 1

2

3∑*=1

V****|z(k(*))|4 +

∑j=1j �=*

V**j j |z(k(*))|2|z(k(j))|2

+6 re(W11123z3(k(1))z(k(2))z(k(3))), (18)

where the interaction coefficients V**j j and W11123 are constants which are derived as part of the calculation. Thetwo horizontal momenta in the space E3 of restricted modes are

I1 = |z(k(1))|2 + 32 (|z(k(2))|2 + |z(k(3))|2), (19)

I2 = 32 (|z(k(2))|2 − |z(k(3))|2), (20)

and it is easy to check that these quantities Poisson commute with each other and with the Hamiltonian HBKS.We repeat the remark that the Hamiltonian HBKS does not include all of the terms of the m = 5, Birkhoff normalform, and that the restricted mode space E3 is not invariant under the full Euler flow Φt , not necessarily under thefifth-order Birkhoff normal form Hamiltonian H(5). Indeed in the deep water case h = +∞ there is a resonantmonomial associated with the quintet P −Q = 3δk(*) − δ−k(*) − δ4k(*) for * = 1, 2, 3, which gives rise to a nonzerocomponent of the Hamiltonian vector field of H(5) orthogonal to E3 [3].

From complex symplectic coordinates of the model equations (11) one introduces initial action–angle variables

z(k(*)) =√I (k(*)) eiθ(k(*)), * = 1, 2, 3, (21)

and we use the vectorial notation I = (I (k(*)))T*=1,2,3, θ = (θ(k(*)))T*=1,2,3. It is useful to introduce rotating

coordinates M = (M1,M2,M3)T, Φ = (Φ1, Φ2, Φ3)

T in a way that respects the two horizontal momenta (19):√11

2M1 = I1 = I (k(1))+ 3

2(I (k(2))+ I (k(3))),

3√2M2 = I2 = 3

2(I (k(2))− I (k(3))).

Define the third action variable to be√

11M3 = 3I (k(1))− I (k(2))− I (k(3)), (22)

in this wayM = RI, whereR is a rotation. SettingΦ = Rθ , the transformation from (I (k(*)), θ(k(*)))3*=1 to (M,Φ)is canonical and the Hamiltonian (18) in the new variables has the form

HBKS =√

112 ω(k

(1))M1 +Q(M)+ 6W11123I3/2(k(1))(M)

√I (k(2))(M)I (k(3))(M) cos(

√11Φ3),

where Q(M) is a quadratic form in M , and the original action variables I (k(*))(M) and the momenta Ij (M) areconsidered as functions of M and are left in this form for convenience in performing calculations. The transformedHamiltonian is independent of the two angle variables (Φ1, Φ2). From [1,21], and also for phenomenologicalreasons, the coefficients of (18) satisfy 3V1111 − V1122 − V1133 = CBKS(1, 3) < 0 and W11123 > 0.

W. Craig / Physica D 152–153 (2001) 434–450 443

Theorem 3. The Hamiltonian system

z(k(*)) = i∂z(k(*))HBKS, * = 1, 2, 3, (23)

is a completely integrable three degrees of freedom system, with Poisson commuting integrals M1,M2 and HBKS.

Define the phase plane S(a) = {(M3, Φ3) : I1(M) = a1, I2(M) = a2}, ignoring the two angles (Φ1, Φ2) (theMarsden–Weinstein reduction). For given (a1, a2) with 0 ≤ |a2| ≤ a1, the set S(a) is topologically a sphere S2,parameterized by {(M3, Φ3) : 0 ≤ Φ3 < 2π/

√11,M−

3 ≤ M3 ≤ M+3 }, with coordinate singularities at the two

poles P* = {I (k(*))(M) = 0}, * = 1, 2. The limitsM±3 , which are affine linear in (M1,M2), result from inspection

of the image of the transformation R, for which the variables I (k(*))3*=1 lie in the positive orthant.Traveling waves. The classical principle is that traveling wave solutions can be characterized as critical points

of the energy HBKS under the constraint of fixed momentum I = (I1(M), I2(M)). Restricting to the phase planeS(a), critical points T of HBKS satisfy the Lagrange multiplier rule

δHBKS = c1δI1 + c2δI2. (24)

In the following discussion, we will take I2 = (3/√

2)M2 ≥ 0, so that I (k(2)) ≥ I (k(3)); since the problem isinvariant under the interchange of k(2) and k(3), the result will be general.

Away from the poles P1, P2 (24) is equivalent to the system of two equations

∂M3HBKS(M3, Φ3) = 0, (25)

∂Φ3HBKS(M3, Φ3) = 0. (26)

The two coordinate singularities P*, * = 1, 2 of the phase plane S(a) will be treated separately. At a solution T 0 =(M0

3 , Φ03 ) of (25) and (26), the two components (c1, c2) of the phase velocity are recovered by the expressions

∂M1HBKS(M03 , Φ

03 ) =

√11

2c1, ∂M2HBKS(M

03 , Φ

03 ) = 3√

2c1,

and clearly HBKS automatically satisfies ∂Φ1HBKS = 0 = ∂Φ2HBKS. The component (26) is explicitly

∂Φ3HBKS = −6√

11W11123I3/2(k(1))(M)

√I (k(2))I (k(3)) sin(

√11Φ3), (27)

and this expression vanishes for√

11Φ3 = mπ , m ∈ Z (and at the poles P1, and P2). Since√

11Φ3 = 3θ(k(1))−θ(k(2))− θ(k(3)) and the angle (θ(k(2))+ θ(k(3))) can be set to 2πj for any j ∈ Z by an appropriate relocation ofthe origin of R2

x , distinct solutions of (27) which are different from the poles P1 and P2 satisfy either√

11Φ3 = 0or π . This corresponds to either θ(k(1)) = 0 or θ(k(1)) = π/3, which are the only two cases.

Theorem 4. Fix the phase plane S(a). If I2(M) > 0, the system of equations (25) and (26) has solutions T on S(a)which are either:

1. at the pole P1 = T1 at which I (k(1))(M) = 0, or2. for angle

√11Φ3 = 0 or π .

In case 2 there are at most two distinct solutions Tj for each of the choices of angle. If I2(M) = 0 there is anadditional solution at the pole T2 = P2 at which I (k(2))(M) = I (k(3))(M) = 0.

Proof. The above discussion of the phase plane has already identified the angles that are possible for (24) to besatisfied, and it has reduced the question to whether the poles P1, and P2 satisfy (24), and to the count of the number

444 W. Craig / Physica D 152–153 (2001) 434–450

of solutions. First of all, ∂Φ3HBKS(P1) = 0, while ∂M3HBKS(P1) is independent of Φ3, and this implies that P1 isan elliptic stationary point for system (23), about which the linearized solutions oscillate with angular frequency∂M3HBKS(P1). This solution is principally composed of Fourier modes for wave numbers k(2) and k(3), with none ofk(1), and it corresponds to the traveling wave solution ηH (x− tc)which in many cases has a characteristic hexagonalspatial structure. It is easy to check (23) in local polar coordinates that when I2(M) > 0, the vector field is smoothand nonstationary in a neighborhood of P2. However, when I2(M) = 0 then P2 is also a stationary point. Thiscorresponds to the two-dimensional solution ηS(x − ct) analogous to the Stokes water wave train, with principalFourier component with wavenumber k(1). We will discuss this solution and its stability properties in more detailbelow.

For stationary points T of HBKS on S(a) away from the poles, then√

11Φ3 = mπ and Eq. (25) reads that

∂M3HBKS = ∂M3Q(M)+ (−1)m∂M3 6√

11W11123I3/2(k(1))(M)

√I (k(2))I (k(3)) = 0. (28)

The function F1(M3) = ∂M3Q(M) is an affine linear function of M3, and solutions of (25) correspond to in-tersections of the graph of the function (−1)m−1F1(M3) with that of F2(M3) = 6

√11W11123∂M3(I

3/2(k(1))(M)√I (k(2))I (k(3))).

Proposition 5. The function F2(M3) is concave over the domain {M3 : 0 ≤ I (k(1))(M), I (k(3))(M)}, F2(M3) >

0 for M3 = M+3 and F2(M3) = 0 for M3 = M−

3 . The latter corresponds to the pole P2 at which I (k(3))

(M) = 0.

Since an affine linear function and a concave function can intersect in at most two points, Theorem 4 follows.

Proof. On the variety I (k(2))(M) = I (k(3))(M), the function F2 simplifies to be

F2 = 6√

11W11123∂M3(I (k(1))3/2I (k(2))) = 6W11123

√I (k(1))( 9

2I (k(2))I (k(1))).

Using that ∂M3I (k(1)) = 3/

√11 and ∂M3I (k

(2)) = −1√

11 = ∂M3I (k(3)), we arrive at the expression

∂2M3F2 = −6W11123

11

1√I (k(1))

3

81

8(I (k(2))+ 2I (k(1))),

which is negative since W11123 > 0. In the general case one can take logarithmic derivatives of the expression

I 3/2(k(1))√I (k(2))I (k(3)) to find that

∂2M3F2 = 6

√11W11123∂

3M3

(I 3/2(k(1))

√I (k(2))I (k(3))

)

= 6√

11W11123

(−567

8

1

I (k(1))3− 5

8

1

I (k(2))3− 5

8

1

I (k(3))3− 45

8

1

I (k(1))I (k(2))2

−45

8

1

I (k(1))I (k(3))2+ 297

8

1

I (k(1))2I (k(2))+ 297

8

1

I (k(1))2I (k(3))+ 5

8

1

I (k(2))I (k(3))2

+5

8

1

I (k(2))2I (k(3))− 27

4

1

I (k(1))I (k(2))I (k(3))

).

The quantity − 58I (k

(2))−3 + 58I (k

(2))−2I (k(3))−1 + 58I (k

(2))−1I (k(3))−2 − 58I (k

(3))−3 within the braces of (29)is factored as −(A−B)2(A+B), which is negative for A = I (k(2))−1 and B = I (k(3))−1 both positive. Hence itscontribution is negative. Inspecting the remaining terms within the braces, the quantity 297I (k(1))−2I (k(2))−1 −

W. Craig / Physica D 152–153 (2001) 434–450 445

45I (k(1))−1I (k(2))−2 − 29790 I (k

(1))−3 is −I (k(1))−1 times a square, as is the analog quantity involving I (k(1)) andI (k(3)), so their contributions to (29) are negative. This leaves (− 567

8 + 2 29790 )I (k

(1))−3 − 274 I (k

(1))−1I (k(2))−1I

(k(3))−1 within the braces, but this is clearly positive, and the claim of concavity in the proposition follows. Theproperties of F2(M3) at the poles P1 and P2 are clear from inspection. �

Stability. Away from the poles P1, and P2, the vector field (23) restricted to the phase plane S(a) is given in thecoordinates (Φ3,M3) by

d

dt

(Φ3

M3

)=(∂M3HBKS

−∂Φ3HBKS

). (29)

The linearized equations of (29) at a stationary point T which is not one of the two coordinate singularities T1 = P1

or T2 = P2 are given by

d

dt

(ϕ3

m3

)=(∂M3∂Φ3HBKS ∂2

M3HBKS

−∂2Φ3HBKS −∂M3∂Φ3HBKS

)(ϕ3

m3

). (30)

Because the poles are coordinate singularities, the stability of the stationary point P1, and of P2 when the transversemomentum I2 = 0 will be studied separately.

From Theorem 4 case 2, a stationary point T can only occur for angles Φ3 for which cos(√

11Φ3) = (−1)m, atan intersection of the graphs of F1(M3) and F2(M3). At such points T , the matrix forming the RHS of (30) takesthe form(

0 ∂M3(F1 ± F2)

±66W11123I3/2(k(1))

√I (k(2))I (k(3)) 0

). (31)

In case m is odd, then T is elliptic (and therefore stable) if 66W11123∂M3(F1 − F2) > 0, and T is hyperbolicin tangent directions to S(a) if this quantity takes the other sign. This is to say that the criterion of stability isfor ∂M3F1 > ∂M3F2, which holds if the graph of F1(M3) crosses that of F2(M3) from below to above as M3

increases.In case m is even, the analogous criterion holds; a stationary point T will be stable if −66W11123∂M3(F1 +

F2) > 0, corresponding to the graph of −F1(M3) crossing that of F2(M3) from below to above, and unstableotherwise.

Without returning to complex symplectic coordinates, it is still possible to understand the nature of the polesP1 = (Φ3,M

−3 ), andP2 = (Φ3,M

+3 )when I (k(2)) = I (k(3)). In particular, if ∂Φ3HBKS(Pj ) = 0 and ∂M3HBKS(Pj )

does not change sign, then Pj is an elliptic stationary point. This is always the case for P1. However, the component∂M3HBKS(Φ3,M

+3 ) could change sign for some values of the momenta I1, and I2, vanishing at angles Φ3 = Φ±

3 .ThenP2 is hyperbolic in a tangent direction to S(a) andΦ±

3 are the angles at which the stable and unstable manifoldsapproach it.

The sequence of bifurcation events. The information contained in the discussion above allows us to sketch asequence of bifurcation events and changes of stability that occur for the BKS model (23). Similar sequences ofbifurcation events occur for any three degrees of freedom system whose Hamiltonian in Birkhoff normal form isexpressed as in (18), where V**j j and W �= 0 are arbitrary.

There are two families of traveling wave solutions of (23) which exist for all values of the parameters. In thephase space S(a) one occurs at the pole T1 = P1, corresponding to a traveling water wave ηH (x − tc) which hasthe form of a hexagonal pattern. Inspecting Eq. (28), a second solution T2 occurs on the line

√11Φ3 = π , for some

value of M3 = M(2)3 , which varies depending upon the momentum parameters (I1, I2). When I2(M) = 0, then

446 W. Craig / Physica D 152–153 (2001) 434–450

T2 = P2 occurs at the second pole, and as a traveling water wave solution it corresponds to the Stokes traveling wavetrain ηS(x − tc). In addition to these, there is a sequence of secondary bifurcations, with concombinant changesof stability, which are described below. We will parameterize the families of solutions with the two momenta(I1, I2).

1. For values of the momentum I1 > 0 which are small, F1 = O(I1), while F2 = OI 3/21 , and the constants

CBKS(1, 3) (which are symplectic invariants) satisfy CBKS(1, 3) < 0. An elementary analysis of the phase planeS(a) (which is best done in a sketch by hand) implies that the only stationary points of system (23) on S(a) occurat T1 = P1 and T2 on the line

√11Φ3 = π . According to the stability criterion of the previous section, both of

these stationary points are elliptic.2. Fixing I2(M) = 0, the pole T2 = P2 remains stable as I1(M) increases, as long as F1(M

+3 ) < F2(M

+3 ). Since

F2(M+3 ) < 0 there is a point I1 = b1 at which F1(M

+3 ) = F2(M

+3 ), beyond which the stationary point T2 = P2

looses stability to a new stationary point T3 = (M(3)3 , π) which moves away from P2 and toward P1. Referring

to the stability criteria for m odd, the new solution T3 is stable, while P2 = T2 now possesses a stable and anunstable manifold in S(a). This new solution corresponds to a genuinely three-dimensional traveling water waveηC(x− tc), having the form of a doubly periodic crescent-shaped wave pattern. Simultaneously, an orbit in S(a)homoclinic to T2 is formed which surrounds the new elliptic stationary point T3. For I1 exceeding b1 by only alittle, the homoclinic orbit passes very close to T3.

On phase planes S(a) with I2 �= 0, the elliptic stationary point T2 moves along the line Φ3 = π withmonotonically decreasing in M(3)

3 .3. In parallel with the event (2), for m even one considers the line

√11Φ3 = 0. For small I1 > 0 the graphs

of −F1(M3) and F2(M3) do not intersect, however, at some point I1 = b2 the two graphs meet at a firstvalue of M−

3 < M(4)3 < M+

3 , subsequent to which a pair of stationary points T4, and T5 is formed. Theone (say, T4) with the smaller value of M3 is hyperbolic, and the other is elliptic, as per the above stabilitycriterion. Depending upon the details of the function F1 and its dependence upon the parameters I1, and I2,either bifurcation b1 or b2 occurs before the other, but the local character of the two bifurcations are independentof this.

From the structure of the nonlinear terms, no further bifurcations occur for the system (23).

Time reversal symmetry. The discrete symmetry (t, η, ξ) �→ (−t, η,−ξ) of the original system (5) is preservedby the BKS model Hamiltonian system with Hamiltonian HBKS. Any solution which exhibits a crescent-shapedelement facing forwards is therefore paired with another solution of the system which has crescent-shaped elementsfacing backwards. In fact with the BKS model the crescent-shaped traveling wave solutions ηC(x − tc) have bothforward and backward facing crescent-shaped elements, and the time reversal transformation maps this solution toitself modulo a spatial translation. This property is the one shortcoming of the model in its ability to describe thethree-dimensional instability of the Stokes wave train ηS(x−tc) and the tendency to form strongly three-dimensionalcrescent shaped and forward facing periodic traveling wave solutions.

The time reversal transformation which takes the model (23) to itself and which fixes the traveling wave solutionT2 involves time reflection and spatial translation. Any traveling wave solution of the model with Hamiltonian (18)has a free surface profile given in the form

ηT (x) = a1 cos(k(1) · x + θ1)+ a2 cos(k(2) · x + θ2)+ a3 cos(k(3) · x + θ3). (32)

Proposition 6. There is a translation x �→ x + α such that the phases can be set to θ2 = 0 = θ3. Suppose thatθ1 = π/3, and consider the reflection x1 �→ −x1. There is a second translation x �→ x + β such that the functionηT (x) is a fixed point of the composition.

W. Craig / Physica D 152–153 (2001) 434–450 447

Proof. Set x = (x1, x2) = (y1 + α1, y2 + α2), and consider its effect on the two phases θ2, and θ3. If αsatisfies( 3

2 k(2)2

32 −k(2)2

)(α1

α2

)=(θ2

θ3

)

then the goal of the first statement is achieved. We note that the determinant of the above transformation is −3k(2)2 �=0. After this first transformation, perform the reflection (x1, x2) �→ (−x1, x2). The phases in the expression (32)become (−k(1)1 x1 +θ1,− 3

2k(1)1 x1 +k(2)2 x2,− 3

2k(1)1 x1 −k(2)2 x2). Setting (x1, x2) = (y1 +2(θ1/k

(1)1 ), y2 −(π/k(2)2 )),

and noting that (32) has been written in cosine series, the phases become (k(1)1 y1 + θ1,32k

(1)1 y1 − k

(2)2 y2 + 3θ1 +

π, 32k

(1)1 y1 + k

(2)2 y2 + 3θ1 −π). For a traveling wave solution for which θ1 is any odd multiple of π/3 and a2 = a3,

this recovers expression (32). In the case of the traveling wave solutions T2, which are the ones which exhibit thestrongly crescent-shaped features reminiscent of the experiments of Su et al. [23], Su [24] and Collard and Caulliez[2], both θ1 = π/3 and a2 = a3. �

5. A new deep water resonant model

The BKS model Hamiltonian involves three Fourier modes, and its character depends upon one fifth-order resonantinteraction between them. The derivation of the model involves an assumption that these three modes are the onlyones of principal importance to the instability of the Stokes wave train to a crescent-shaped wave, and in the modelthe amplitudes of all other Fourier modes are set to zero. However, in the deep water problem, with h = +∞, thenω2(k) = g|k| and there is another prominent fifth-order resonance relation that might be suspected of playing a role.Specifically, we include the Fourier mode with wave number k(4) = 4k(1), which adds another resonant quintet

3ω(k(1))− ω(−k(1))− ω(k(4)) = 0 (33)

to the reduced system. The additional resonant interaction is arguably quite important to the dynamics, especially inconsidering water wave solutions close to the Stokes wave train, as they involve a principal component of the solutionin the mode of wave number k(1). The model taking this into account is derived under the same considerations atthat of the BKS model, however, involving the four Fourier modes with wave numbers k(1), k(2), k(3), and k(4). Thewater wave Hamiltonian is transformed to Birkhoff normal form to fifth order, and then restricted to the subspace{z : z(k) = 0 for all k �= k(j), j = 1, . . . , 4}, and it takes the form

HR =4∑*=1

ω(k(j))|z(k(j))|2 + 1

2

4∑*=1

V����|z(k(*))|4 + 34∑*=1

∑j �=*

V**j j |z(k(j))|2|z(k(*))|2

+6W11123 re(z(k(1))3z(k(2))z(k(3))+W11114 re z(k(1))3z(k(1))z(k(4)). (34)

The two components of the horizontal momentum are

I1 = I (k(1))+ 32 (I (k

(2))+ I (k(3)))+ 2I (k(4)), I2 = 32 (I (k

(2))− I (k(3))), (35)

where |z(k(j))|2 = I (k(j)), and these are conserved quantities under the time evolution

∂t z(k(j)) = i∂z(k(j))HR(z). (36)

To put this system into rotating coordinates, perform the canonical transformation

Ψ = Rθ, N = (RT)−1I, (37)

448 W. Craig / Physica D 152–153 (2001) 434–450

where R is given by

R =

√2

19

3

2

√2

19

3

2

√2

192

√2

19

01√2

− 1√2

0

3√11

− 1√11

− 1√11

0

2√5

0 0 − 1√5

, (38)

then√

11Ψ3 = 3θ(k(1))− θ(k(2))− θ(k(3)) and√

5Ψ3 = 2θ(k(1))− θ(k(4)). The Hamiltonian HR is transformedto

HR =4∑*=1

ω(k(*))I (k(*))+ 1

2

4∑*=1

V����I (k(*))2 + 3

4∑*=1

∑j �=*

V**j j I (k(j))I (k(*))

+6W11123I (k(1))3/2

√I (k(2))I (k(3)) cos(

√11Ψ3)+W11114I (k

(1))2√I (k(4)) cos(

√5Ψ4). (39)

The action variables I (k(j)) are to be considered as functions of N . We observe that HR is independent of thetwo angles Ψ1, and Ψ2, corresponding to the two integrals of motion I1, and I2, however, the Hamiltonian is notintegrable in any obvious way. As with the BKS model, the reduced phase space is given by S(a) = {(N,Φ) : I1 =a1, I2 = a2}, which is four-dimensional. Traveling wave solutions with momenta Ij = aj , j = 1, 2 correspond tostationary points of HR on S(a), with the components of its phase velocity of a solution T given by

∂N1HR(T ) =√

192 c1, ∂N2HR(T ) = 1

3

√2c2. (40)

Proposition 7. On the reduced phase space S(a), stationary points of HR occur either for

1.√

11Ψ3 = 3θ(k(1))− θ(k(2))− θ(k(3)) = m1π ,2.

√5Ψ4 = 2θ(k(1))− θ(k(4)) = m2π

for some integers m1, and m2, or else for I (k(1))(N) = 0. The latter case corresponds to a traveling wave solutionηH (x−tc) exhibiting a hexagonal spatial pattern. If the transverse momentum I2 = 0, then there is a third possibility,in which the set {I (k(2)) = 0 = I (k(3))} contains a stationary point of HR on S(a). This corresponds to the Stokeswave train solution ηS(x − tc).

Proof. Stationary points which are not at singularities of the symplectic polar coordinates must satisfy

0 = ∂Ψ3HR = −6√

11W11123I (k(1))3/2

√I (k(2))I (k(3)) sin(

√11Ψ3),

0 = ∂Ψ4HR =√

5W11114I (k(1))2

√I (k(4)) sin(

√5Ψ4). (41)

The effect is that (2) holds, which for each m2 ∈ Z fixes Ψ4 as a function of Ψ3. Furthermore (1) holds, setting Ψ3

to discrete values similar to the critical points of the BKS Hamiltonian. Upon inspection, I (k(1)) = 0 is a stationarypoint of HR on S(a) at the pole P1, but I (k(4)) = 0 only occurs as a coordinate singularity. �

W. Craig / Physica D 152–153 (2001) 434–450 449

Critical points of (39) which do not occur at one of the poles will satisfy (41) as well as

0 = ∂N3HR = ∂N3B(I (N))

+ 6√11W11123

9

2

√I (k(1))I (k(2))I (k(3))− 1

2I (k(1))3/2

√I (k(3))

I (k(2))+ I (k(2))

I (k(3))

cos(

√11Ψ3)

+ 6√11W11114I (k

(1))√I (k(4)) cos(

√5Ψ4), (42)

0 = ∂N4HR = ∂N4B(I (N))+ 18√5W11123

√I (k(1))I (k(2))I (k(3)) cos(

√11Ψ3)

+ 1√5W11114

(4I (k(1))

√I (k(4))− 1

2I (k(1))2

√I (k(4))

−1)

cos(√

5Ψ4), (43)

whereB(I (N)) is the quadratic form in the action variables which appears in (39). The coefficients of the HamiltonianHR in (39) are taken to satisfy the structure conditionsW11123 > 0,W11114 > 0,CR(1, 3) = 3V1111−V1122−V1133 <

0, and CR(1, 4) = √22V1111 − V1144 < 0. These are consistent with the BKS model and [3,6].

Given a critical point T of (39) on S(a) which is not at a pole, and given the result of Proposition 7 a stabilitycriterion for the system is straightforward to work out.

Proposition 8. Suppose that a stationary point T of the system with Hamiltonian (39) on S(a) is not one of thepoles. Then T is stable if both

∂2N3HR∂

2Ψ3HR + ∂2

N4HR∂

2Ψ4HR > 0, (44)

∂2Ψ3HR∂

2Ψ4HR(∂

2N3HR∂

2N4HR − (∂N3∂N4HR)

2) > 0. (45)

The sequence of bifurcation events. There is now a clear sequence of bifurcation events for traveling wave solutionsof system given by (39). This is very closely related to the sequence of bifurcation events for the BKS system.

1. For small x1-momentum I1, each phase plane S(a) contains only two critical points of HR; one which is at thepole T1 = {I (k(1)) = 0}, and the other T2 close to the pole {I (k(3)) = 0} (we are again normalizing so that 0 <I (k(3)) ≤ I (k(2)). When I (k(3)) = I (k(2)) then this critical point T2 occurs at the pole {I (k(2)) = I (k(3)) = 0}itself.

2. At some bifurcation point I1 = c1 > b1, on the phase plane S(a) such that I (k(2)) = I (k(3)), the stationary pointT2 loses stability in a Hamiltonian saddle-node bifurcation. A new critical point T3 emerges on the phase planeS(a) for which

√11Ψ3 = π and I (k(2)) = I (k(3)) > 0. These solutions ηC(x−tc) have three-dimensional struc-

ture. In particular, they feature the phase shift θ1 = π/3 giving them the characteristic feature of crescent-shapedwater waves. According to the stability criterion of Proposition 8, the solution T3 is stable.

3. At a separate bifurcation point I1 = c2 > b2, a pair of stationary points T4 and T5 is created on the set{√11Ψ3 = 0}. At these points all action variables I (k(*)), * = 1, . . . , 4 are nonzero. One of these critical pointsis stable, while the other is very likely to be unstable (it depends in general on the parameters).

Further bifurcations can occur for larger values of the momentum I1, but these are less relevant to the water wavesproblem as they take place further from the weakly nonlinear regime under which this model is derived, and wewill not describe the resulting stationary points here.

450 W. Craig / Physica D 152–153 (2001) 434–450

Time reversal symmetry. The time reversal symmetry (t, η, ξ) �→ (−t, η,−ξ) leaves the R-system invariant.Similar to the BKS model, time reflection composed with a specific translation maps the candidate crescent-shapedsolutions ηC(x − tc) to themselves. It remains an open problem whether a Hamiltonian model of water waves willexhibit solutions which satisfy all three of the criteria for nonlinear crescent-shaped solutions that are specified in[21], which are closer to the desired unidirectional crescent-shaped waveforms of the wave tank experiments of Suet al. [23] and Su [24].

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